PREDATORS-PREY MODELS WITH COMPETITION PART I: EXISTENCE, BIFURCATION AND QUALITATIVE PROPERTIES

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1 PREDATORS-PREY MODELS WITH COMPETITION PART I: EXISTENCE, BIFURCATION AND QUALITATIVE PROPERTIES HENRI BERESTYCKI AND ALESSANDRO ZILIO Abstract. We study a mathematical model of environments populated by both preys and predators, with the possibility for predators to actively compete for the territory. For this model we study existence and uniqueness of solutions, and their asymptotic properties in time, showing that the solutions have different behavior depending on the choice of the parameters. We also construct heterogeneous stationary solutions and study the limits of strong competition and abundant resources. We then use these information to study some properties such as the existence of solutions that maximize the total population of predators. We prove that in some regimes the optimal solution for the size of the total population contains two or more groups of competing predators. 1. Introduction Systems of reaction-diffusion equations are ubiquitous in mathematical biology, as they serve as a basic framework for modeling a diversity of biological and ecological mechanisms. In particular, the study of population dynamics often involves such systems. In a recent paper [4], we have introduced a new reaction-diffusion system to describe the emergence of territoriality in predatory animals. As a matter of fact, this question, to a large extent, remains puzzling. Specifically, our aim was to understand whether selfish and non organized behaviors of predators are sufficient mechanisms to explain the emergence of territoriality. In this model, we only consider basic mechanisms that characterize an environment inhabited by predators and preys. Given a region R n with n 2 in practice (the restriction on the dimension is purely for modeling reasons occupied by N + 1 groups of animals. The density of the first one, which we denote as u, of preys while the remaining N densities, denoted 21 Mathematics Subject Classification. Primary: 35Q92; secondary: 35A1, 35B25, 35B32, 92D5. Key words and phrases. systems of parabolic equations, asymptotic analysis, stability of solutions, bifurcation analysis, non-constant solutions, competition, optimization. 1

2 2 HENRI BERESTYCKI AND ALESSANDRO ZILIO by w 1,..., w N, of predators. Each of the densities evolves in time following a typical law of Lotka-Volterra type. The model proposed in [4] is synthesized in the form of the system w i,t d i w i = ( ω i + k i u µ i w i β j =i a ij w j w i (1.1 u t D u = (λ µu i=1 N k iw i u for (x, t (, +, completed by homogeneous Neumann boundary and smooth initial conditions. The parameters of the model are easily explained: some terms in the equations model internal mechanism in the populations D, d 1,..., d N are the diffusion coefficients of the different populations, and thus are always considered positive in the following; λ > is the effective reproduction rate of the preys, and µ stands for the possible saturability of the environment due to an excess of preys; ω i is the mortality coefficient of the group i of predators that takes into account the starvation caused by the absence of the prey u, and µ i takes into account possible saturation phenomena in the predator populations (for instance, an internal low level competition between member of the same density. The other terms are, on the other hand, responsible for the interaction between different densities k 1,..., k N govern the predation rates. That is, k i is the success of predator w i in catching prey u as a factor of the probability of an encounter; the elements a ij > represent how the presence of the density w j affects the density w i, and the particular choice of the sign suggest that we only consider competing interactions. The parameter β on the other hand expresses the strength of the interaction: the higher the value of β, the more aggressive is the behavior of predators between different groups. Similar models have already been introduced in the ecological and mathematical literature, starting from the seminal paper by Volterra [25] on predator-preys interactions, to the more recent contribution [13] that started the study of strongly interaction systems of elliptic equations (in that case, modeling populations of competing predators without preys, the study

3 PREY-PREDATORS, PART I 3 on the evolution of dispersal by means of systems of many interacting predators [15], and the papers on the qualitative properties of the solutions to such systems [7, 6, 14]. The novelty in our model, that complicates the analysis but allows for more profound results, is the inclusion in the system of the equation for the preys and of the competition between the predators. A more in depth comparison with the results in the scientific literature can be found in [4], to which we refer the interested reader for more detailed biological considerations. The main results in this paper regarding model (1.1 are summarized as follows. In the following sections we will give more general statements. For sufficiently smooth and positive initial data, the system (1.1 admits a unique, bounded and smooth solution, defined for all t (see Lemma 2.1. The competition is a driving force in the heterogeneity of the set of solutions, indeed the set of stationary solutions of the system collapse to the set of constant solutions if β is small (see Proposition 2.2, and is very rich for β large (see for instance Theorems 3.9, 3.15 and 5.1, but under some assumptions the asymptotic behavior β + can be described accurately (see Lemma 2.1 and Proposition 2.2. The solutions of the stationary system are regular, independently of the strength of the competition, and they converge to segregated configurations when β + (see Proposition 4.1 in which territories of different competing groups do not overlap. This allows us to define also limit solutions to (1.1 in the case β = +. Aggressiveness (β 1 may help to resist an invasion by a foreign group (see Propositions 2.4 and 2.8. On the other hand, the strong competition limits the number of different groups that can survive in a given domain. Indeed we have Theorem. For a given smooth domain R n, there exist N N and β > such that if β > β and w β = (w 1,β,..., w N,β, u β is a solution to (1.1 then either at most N components of (w 1,β,..., w N,β are strictly positive and the others are zero; or the solution is such that (w 1,β,..., w N,β C,α ( + u β λ/µ C 2,α ( = o β(1 for every α (, 1.

4 4 HENRI BERESTYCKI AND ALESSANDRO ZILIO Furthermore, the threshold value N has the following upper-bound: N 4π if n = 2, and similar estimates hold in any dimension. max λk i µω i i=1,...,k d i µ In the theorem A B stands for A B + o(b where o(b/b as B +. This theorem states that, when β is sufficiently large, solutions to (1.1 are either close to constant (and small solutions, or they have at most N (+1 non trivial components. This result has important repercussions on the biological interpretations of the model, as it imposes an upper-bound on the total number of hostile groups of predators that can survive in a given environment. Moreover, the upper-bound itself has important ecological consequences: We have explored them in [4]. Finally, under some assumptions on the coefficients, there exist a number of densities of predators N N and a solution (w 1,..., w N, u of (1.1 that maximize the total population of predators. We show furthermore that, in many cases, this optimal configuration is given by two or more densities of predators that have very aggressive behavior between each other, rather than by a simple homogeneous group that displays no aggressiveness between its components. Theorem (see Theorems 6.5 and 7.4. For any given smooth domain, there exist a number N N and a solution (w 1,..., w N, u of (1.1 at most N + 1 non trivial components (possibly with β = + that maximizes the functional P(w 1,..., w N, u = N w i i=1 among the set of all non negative solutions of (1.1. Moreover, if is a rectangular domain (in any dimension and µ is sufficiently small, the maximum is attained by a solution with two or more densities of predators, that is, N 2 and β >. An interesting open problem remains for the second conclusion of the theorem, concerning the case when N 2. Indeed, we believe that the result holds true for rather general domains, and numerical simulations sustain our claim, but the proof is so far elusive. A consequence of the theorem is that competition between predators can be beneficial not only for the preys, but for the total population of predators as well.

5 PREY-PREDATORS, PART I 5 Structure of the paper. The paper is organized as follows: in Section 2 we consider some basic properties of the system, such as existence and regularity of solutions, together with some asymptotic properties of the system, focusing on the stability properties of specific solutions. In Section 3, thanks to a bifurcation analysis, we show that the set of stationary solutions is very rich. In Section 4, we present some uniform estimates that we later use to give a precise description of the solutions for large competition. In Section 5, we show a more precise description of the bifurcation diagram in dimension one. In Section 6, we investigate some properties of the system with a large number of components, and finally, in Section 7, we show configurations in which the maximizers of the integral of the densities w i are spatially heterogeneous. The interested reader will find a companion paper [4] to this one, where we investigate the biological and ecological interpretations of the mathematical results herein contained. The present paper is the first of a series in which we investigate properties of the system (1.1. In a second part [1] we give deeper a priori estimates of the solution to the elliptic counterpart of system (1.1. We exploit these properties in [2] to give a more precise description of the set of solutions. Finally, in [3] we prove results regarding the parabolic version of the model. 2. Basic properties of the solutions In this section we investigate some basic properties of the system. First, we establish existence and uniqueness results for the solutions. Then we analyze the long time behavior of the set of solutions. We also consider stability properties of a special class of solutions, namely those with only one predator and one prey, that is where all the w i are zero but one. We recall that the system reads: (2.1a w i,t d i w i = ( ω i + k i u µ i w i β j =i a ij w j u t D u = (λ µu i=1 N k iw i u w i

6 6 HENRI BERESTYCKI AND ALESSANDRO ZILIO in a domain Q := (,, with R n open, smooth, bounded and connected. It is completed by boundary and smooth initial conditions ν w i = ν u = on (, + (2.1b w i (x, = wi (x, u(x, = u (x on {}, where ν denotes the unit outward normal vector field on. We start with the following existence result Lemma 2.1. Let (w 1,..., w N, u C,α ( be a non-negative initial condition for the system (2.1. There exists a unique solution (w 1,..., w N, u C 2,α x C 1,α/2 t (Q for all α (, 1 which is defined globally for all t >. Moreover the solution is bounded in L (Q and for any ε > there exists T ε > such that sup w i (x, t λk i µω i + ε (x,t [T ε,+ µµ i sup u(x, t λ (x,t [T ε,+ µ + ε. Consequently, if there exists and index i {1,..., N} such that λk i µω i, then lim sup w i (x, t =. t + In view of the last property, we shall also assume in the following that (H the relation λk i > µω i holds for all i = 1,..., N. x Proof. The existence of solution for t [, t ] with t > small follows from standard arguments, since the semi-linear terms of the system are locally Lipschitz continuous: in order to extend the existence result for all time t >, it suffices to show an a priori L uniform bound on the solutions. First of all, we can observe that each single equation of the system (2.1 is satisfied by the trivial solution (u = and w i = for some i. Consequently, the comparison principle applied to each equation implies that the solutions, when defined, are strictly positive for positive t. Using this information, we focus our attention on the equation satisfied by the density u, that

7 PREY-PREDATORS, PART I 7 is (2.2 u t D u = u(x, = u (x. (λ µu N i=1 k iw i u Let U C 1 (R + be the solution of the initial value problem U = λu µu 2 for t > U( = max{λ/µ, sup x u (x} >. The family of solution U is decreasing in t > and U(t λ/µ as t + : as a result, for any ε > there exists T ε finite such that U(t λ/µ + ε for any t T ε. Clearly, since each w i is non-negative, u(x, t U(t for all x. Therefore, u(x, t is then bounded uniformly. Taking into account this information, we see that each w i satisfies (2.3 w i,t d i w i w i (x, = wi (x. ( ω i + k i U µ i w i β j =i a ij w j w i Using a similar reasoning as before, we can introduce the auxiliary function W i C 1 (R + solution to the initial value problem Ẇ i = ( ω i + k i U µ i W i W i for t > { } W i ( = λki µω max i µµ, sup i x wi (x >. Clearly W i is uniformly bounded in t and moreover, W i (t (λk i µω i /(µµ i as t +. Hence, we deduce that for any ε > there exists T ε finite such that W i (t (λk i µω i /(µµ i + ε for any t T ε. Again, since each w i is non negative and u U, we see that W i is a super-solution for (2.3 and thus w i is bounded uniformly. The previous uniform upper bounds are enough to ensure that the solution can be extended for all time t > and also yield the asymptotic estimates. Before going further we recall a result in [9] about the asymptotic behavior in time of solutions to systems of reaction diffusion equations. We let L be the Lipschitz constant of the

8 8 HENRI BERESTYCKI AND ALESSANDRO ZILIO semi-linear term in (1.1 on the invariant region of Lemma 2.1. That is, letting ( ω i + k i S µ i s i β j =i a ij s j s i F(s 1,..., s N, S = (λ µs i=1 N k is i S we define { L = sup F(s 1,..., s N, S ; < s i < λk i µω i, < S < λ }. µµ i µ We observe that, thanks to the assumptions, L is finite and positive. We also let d = min{d 1,..., d N, D} and define γ 1 to be the first non trivial (that is, positive eigenvalue of the Laplace operator in with homogeneous boundary conditions. Finally, for any solution (w 1,..., w N, u of (1.1, we let w i (t = 1 w i (x, tdx, ū(t = 1 u(x, tdx. Applying [9, Theorem 3.1] to our system (1.1 we have the following result on the asymptotic behavior of the solutions for large time. Proposition 2.2. Let σ = dγ 1 L. If σ >, then for any non negative initial condition (w 1,..., w N, u C,α (, the corresponding unique solution of the system (2.1 converges exponential towards spatially homogeneous solutions, that is, for any < σ < σ there exists a constant C > such that N w i L 2 ( + u L 2 ( Ce σ t i=1 N w i (, t w i (t L ( + u(, t ū(t L ( Ce σ t/n. i=1 Moreover, the vector ( w 1,..., w M, ū is a solution of a the system of ordinary differential equations of the form w i = ( ω i + k i ū µ i w i β j =i a ij w j ( ū t = λ µū i=1 N k i w i ū + g(t w i + g i (t

9 PREY-PREDATORS, PART I 9 with and w i ( = 1 w i (x dx, ū( = 1 N g i (t + g(t Ce σ t. i=1 u(x dx. Proof. The proof is a straightforward application of [9, Theorem 3.1]. We only observe that by Lemma 2.1 we know that from any positive initial data and any ε > there exists T ε > such that the corresponding unique solution is contained in the region { < w i (x, t < λk i µω i + ε, < u(x, t < λ } + ε, x µµ i µ for all t T ε. Now, if σ >, by regularity of F for any ε > sufficiently small { σ = dγ 1 sup F(s 1,..., s N, S ; < s i < λk i µω i + ε, < S < λ } µµ i µ + ε > and we can apply [9, Theorem 3.1] to obtain the stated exponential estimates. The important consequence of the previous proposition is that the behavior of the solutions, in the regime σ > is well described by the corresponding system of ordinary differential equations. It also gives us a complete characterization of the set of stationary solutions of (2.1, which is then given by the (spatially constant solutions of F(w 1,..., w N, u =. For instance, by studying the stability of the stationary and homogeneous solutions (see Proposition 2.4 and Lemma 3.3 below we will see that in this case, when β > the only stable stationary solutions are those that have u > and only one component of (w 1,..., w N non trivial (and positive. Finally, we observe that the condition σ > can be violated in three different ways: (i lowering the diffusion coefficients, (ii enlarging the domain or (iii augmenting the Lipschitz constant L. This last possibility, which is the one that we mainly explore later, can be result for instance form taking µ small and β large enough. We now start investigating the equilibria of the system, in particular we want to analyze what is the impact of the competition parameter on the possible heterogeneity of the solutions of the system.

10 1 HENRI BERESTYCKI AND ALESSANDRO ZILIO We first recall the well known result by Dockery et al. [15] on a related simpler model (2.4 w i,t d i w i = ν w i = ( a(x N j=1 w j w i in (, + on (, +. This system describes N populations that share the same spatially distributed resource a. These populations do not compete actively against each other, but they do suffer from overpopulation, which is model by the logistic term in the equations. Here a is a smooth non constant function such that the principal eigenvalue of each of the elliptic operators d i w = aw + λw ν w = in on, denoted by λ(d i, a, is strictly negative (implying, in particular, the instability of the zero solution. Exploiting the particular symmetric structure of the interaction/competition term, Dockery et al. were able to show that if a is not constant, the only asymptotically stable equilibrium of the system is the stationary solution that has all the components w i zero except for the one with the smallest diffusion coefficient d i. Moreover, the same result holds if we introduce a small mutation term in the system, which in terms imply also an evolutionary advantage for small diffusion rates. The classic interpretation of this result is that, since the densities w i in (2.4 are equivalent if not for the diffusion rates, the density which can concentrate more on favorable zones (maxima of a will benefit more than the others and will end up eliminating them. In what follows, we shall show that this is not the case for the model we are considering, and in particular we prove that for β sufficiently large, all the solutions that have only one nontrivial density of predators are asymptotically stable. Remark 2.3. In order to justify the link between the model (2.4 and our model (2.1, let us consider the limit case of (2.1 in which the density u has a very fast dynamic with respect to the other components, that is, let us assume that for each t >, the density u reaches

11 PREY-PREDATORS, PART I 11 instantaneously its non-trivial inviscid equilibrium state, λu µu 2 u N i=1 k i w i = = u = 1 µ ( λ N i=1 k i w i Substituting the previous identity in the equations satisfied by w i we obtain ( k w i,t d i w i = i λ µ ω i k ( i µ w i βa ij + k i w µ j w i j =i In the simplified case k i = µ, ω i = ω and β =, we obtain finally ( ( w i,t d i w i = λ ω N j=1 w j w i = a N j=1. w j w i. We thus obtain the model of Dockery et al. [15] with a = λ ω. Notice that we could consider that λ and ω depend on the location in space (certain locations being more favorable than others. More details can be found in [4]. We have the following Proposition 2.4. For a fixed i {1,..., N}, let W be the stationary solution of (1.1 which has only the i-th densities of predator which is nontrivial. Then v is constant and v = (,..., w i,...,, ũ with w i = λk i µω i k 2, ũ = ω i. k i i There exists β such that if β β then W is asymptotically stable with respect to perturbations in C 2,α (. More explicitly, this holds whenever β satisfies the system of inequalities β k ( j ω i ω j j = i. a ji w i k i k j Proof. First of all, by [17, Theorem 1] and [1], we have that the only solution of the system with only the i-th density of predator non trivial is the constant solution W. The study of the stability of this solution is based on a simple analysis of the linearized system around it: we consider the Gâteaux differential around v of the operator describing the system, which is given by d i w i k i w i u + β w i j =i a ij w j [ ( L(v[w 1,..., w N, u] = ωj ] d j w j + k j k ω i j k + β w i i a ji w j for j = i D u + µ ω i k u + ω i i w i

12 12 HENRI BERESTYCKI AND ALESSANDRO ZILIO for all (w 1,..., w N, u C 2,α ( with homogenous Neumann boundary conditions. To ensure the stability of the solution we need to show that the spectrum of L is contained in C + = {z C : Re(z > }, that is for any (w 1,..., w N, u = and γ C L(v[w 1,..., w N, u] = γ(w 1,..., w N, u = Re(γ >. In the previous system, the components corresponding to j = i are decoupled from the others, and thus their presence does not influence the stability of W. This solution W is stable if and only if ( ωj k j ω i + β w k j k i a ji > j = i i which gives the condition established by the proposition; indeed, under this assumption the components w j with j = i are necessarily trivial. Let us show that this condition is enough to ensure the stability: we suppose that the previous system of inequalities holds but there exist (w 1,..., w N, u = and γ C with Re(γ solution to L(v[w 1,..., w N, u] = γ(w 1,..., w N, u. Then necessarily w j = for all j = i, and the system is reduced to (2.5 d i w i = γw i + k i w i u ( D u = ω i w i + γ µ ω i k u i ν w i = ν u = on Since any weak solution to the previous system is regular, the stability in C 2,α ( can be deduced from the solvability of the system in H 1 (. To analyze it, let {(γ h, ψ h } h N be the spectral resolution of the Laplace operator with homogeneous Neumann boundary condition in (let us recall that γ = and γ h > for h > ; since {ψ h } h N is a complete basis of L 2 (, we can write w i = a h ψ h and u = b h ψ h h= h= as series converging in L 2 (. Inserting these relations in (2.5 and using the orthogonality of the eigenfunctions, we see that the linear system (2.5 is equivalent to the sequence of algebraic

13 PREY-PREDATORS, PART I 13 eigenvalue problems d i γ h a h k i w i b h = γa h ( for h N. Dγ h + µ ω i k b i h + ω i a h = γb h By direct inspection, we can observe that a h = if and only if b h =. Thus, solving the first equation in b h and substituting the result in the second, we find that γ must be a solution to ( Dγ h + µ ω i γ (d k i γ h γ + k i w i ω i = i that is γ = 1 ( 2 (D + d i γ h + µ ω i k i ± ( (D + diγh + µ ω 2 i 4k k i w i ω i i and in particular Re(γ >. We observe that the diffusion rates do not play any role in the stability of the solutions, while a crucial role is played by the ratio ω i /k i. In particular if i is such that ω i k i < ω j k j j = i then the solution W is asymptotically stable also in a slightly cooperative environment, that is for β < and small in absolute value. This is a consequence of the fact that the semi-trivial solutions are constant and the different diffusion rates do not play a direct role in the stability of the solution (that is, advantage of low/high diffusion rate. In this setting, the quantity ω i /k i can be interpreted as the fitness of the i-th population. One could then wonder whether the previous stability result is a spurious consequence either of the fact that the simple solutions are constant or of another specific feature of this particular formulation of the system. To clarify this issue, we shall now adapt the proof to a very general framework. Let us consider the following operator [ ] L i w i f i (x, u, w i β j =i g ij (x, w i, w j w i for all i {1,..., N} S β (v := Lu f (x, u, w 1,..., w N u defined for v = (w 1,..., w N, u in the set F( = { } v C 2,α (; R N+1 : νi w i = ν u = on

14 14 HENRI BERESTYCKI AND ALESSANDRO ZILIO where the respective operators L i and L stand for linear strongly elliptic operators of the form L i w i = div(a i (x w i, Lu = div(a(x u associated with some smooth and uniformly elliptic symmetric matrices A i and A, and theν i and ν denote the co-normal vector fields associated to the corresponding elliptic operators. We assume in the following that all the terms in the operator S β are sufficiently smooth to justify the following computations, and moreover we suppose that there exists positive constants C such that for any v F( of non negative components we have f i (x, u, w i C(1 + u w i f (x, u, w 1,..., w N C(1 u g ij (x, w i, w j Based on the previous notation, a function v F( is a solution of the generalized model if S β (v = while a function v C 1 (R + ; F( C(R + ; F( is a solution to the parabolic model if t v + S β (v = t > v( = v v F(. Using the previous assumptions, we have Lemma 2.5. For any non-negative initial datum v F( there exists a unique solution v of the previous parabolic problem. Moreover, there exists T > and M >, independent of β, such that w 1 (t, x,..., w N (t, x, u(t, x M for all t T, x. If there exist i {1,..., N}, t > and x such that w i (t, x = (respectively, u(t, x =, then w i (respectively, u. Proof. The proof follows directly from the maximum principle, and thus we omit it (see Lemma 2.1 for reasoning of this kind.

15 PREY-PREDATORS, PART I 15 In an analogous fashion, we have a corresponding result for the stationary model. Among the class of all possible solutions, we are interested in the particular case of solutions that have only one component among the first N which is non-trivial. Definition 2.6. For a given i {1,..., N}, a solution v F( is said to be i-simple if w j for all j = i and the other components are positive. Let us observe that if v F( is an i-simple solution for S β, then it is an i-simple solution for any value of β. For a given solution v F(, let L(v be the Gâteaux derivatives of S β in F(, that is for any ϕ F(: S β (v + εϕ S β (v L(v[ϕ] = lim ε ε [ ] L i ϕ i f i (x, u, w i β j =i g ij (x, w i, w j ϕ i = f i,u (x, u, w i w i ϕ f i,wi (x, u, w i w i ϕ i +β j =i g ij,wi (x, w i, w j w i ϕ i + β j =i g ij,wj (x, w i, w j w i ϕ j Lϕ f (x, u, w 1,..., w N ϕ f,u (x, u, w 1,..., w N ϕ N i=1 f,w i (x, u, w 1,..., w N ϕ i Analogously, for any fixed i {1,..., N} we define the i-th partial derivatives L i (v as the Gâteaux derivatives of S β in F( with respect to the direction ϕ F( such that ϕ = (,..., ϕ i,...,, ϕ, that is L i ϕ i [ ] f i (x, u, w i β j =i g ij (x, w i, w j ϕ i f i,u (x, u, w i w i ϕ f i,wi (x, u, w i w i ϕ i +β j =i g ij,wi (x, w i, w j w i ϕ i L i (v[ϕ] = for j = i Lϕ f (x, u, w 1,..., w N ϕ f,u (x, u, w 1,..., w N ϕ f,wi (x, u, w 1,..., w N ϕ i

16 16 HENRI BERESTYCKI AND ALESSANDRO ZILIO Accordingly, we recall that a solution v F( is (strongly stable if any non-trivial solution (γ,ϕ of L(v[ϕ] = γϕ has necessarily Re(γ >. For i-simple solutions we have Definition 2.7. For a given i {1,..., N}, an i-simple solution v F( is one-predator stable if any non-trivial solution (γ, ϕ of L i (v[ϕ] = γϕ with ϕ = (,..., ϕ i,...,, ϕ has necessarily Re(γ >. An i-simple solution is thus one-predator stable if it is stable with respect to all the admissible perturbations that leave unchanged the zero components w j for j = i. Clearly, if an i-simple solution is stable it is also one-predator stable: under suitable conditions, the inverse is true. Proposition 2.8. For a given i {1,..., N}, let us assume that inf g ji(x,, s > for all s > and j = i. x If v F( is an i-simple one-predator stable solution v F(, then there exists β such that v is a stable solution for all β > β. Proof. The i-simple solution v = (,..., w i,...,, u is stable if L i ϕ i [ ] f i (x, u, w i β j =i g ij (x, w i, ϕ i f i,u (x, u, w i w i ϕ f i,wi (x, u, w i w i ϕ i +β j =i g ij,wi (x, w i, w i ϕ i + β j =i g ij,wj (x, w i, w i ϕ j = λϕ i L j ϕ j [ f j (x, u, βg ji (x,, w i ] ϕ j = λϕ j Lϕ f (x, u,,..., w i,..., ϕ f,u (x, u,,..., w i,..., ϕ N i=1 f,w i (x, u,,..., w i,..., ϕ i = λϕ

17 PREY-PREDATORS, PART I 17 has a nontrivial solution ϕ F( if and only if Re(λ >. Let us consider the equations of index j = i, which are decoupled from the other equations in the system. They read: [ ] L j ϕ j = f j (x, u, βg jh (x,, w i + λ ϕ j in L j ν ϕ j = on Since the operator L j is self-adjoint 1, any non-trivial solution of the system must have λ R. The solution v being an i-simple solution, by the maximum principle it follows that inf w i(x = c >. x As a result, thanks to our assumptions, there exists β such that f j (x, u, β sup x g ji (x,, w i for all j = i. Choosing β > β and testing the equation in ϕ j by ϕ j itself, we obtain A j (x w j w j = [ fj (x, u, βg ji (x,, w i + λ ] ϕ 2 j < λ ϕ 2 j thus either λ > or the component ϕ j =. On the other hand, assuming that Re(λ, we find a contradiction with the internal stability of the solution v. 3. Existence of non homogeneous solutions: a bifurcation analysis We continue the investigation of the asymptotic properties of the system (1.1, by now studying the set of solutions of the corresponding (stationary elliptic problem. We consider here the model (1.1 under the assumption that the domain is occupied by only two groups of predators, having the same parameters. This system reads: d w 1 = ( ω + ku βw 2 w 1 d w 2 = ( ω + ku βw 1 w 2 D u = (λ µu k(w 1 + w 2 u ν w i = ν u = in in in on 1 More precisely, the operator is self-adjoint if seen as an operator acting on H 1 ( functions, and the conclusion can be reached using the regularity assumptions on its coefficients.

18 18 HENRI BERESTYCKI AND ALESSANDRO ZILIO for which we look for solutions (w 1, w 2, u C 2,α (. Let us point out that here we take µ 1 = µ 2 =. Alternatively, we can easily generalize the results that will we show in the following to the case of positive saturation coefficients (though the computations are inevitably more involved. Since we are looking for stationary solutions, the system can be simplified by some linear substitutions. Indeed, letting u d D D2 u, λ λd, µ µ, k kd, ω ωd, β βd d we can reformulate the system as w 1 = ( ω + ku βw 2 w 1 in (3.1 w 2 = ( ω + ku βw 1 w 2 u = (λ µu k(w 1 + w 2 u ν w i = ν u = in in on We recall the definition of the set F( := { } (w 1, w 2, u C 2,α ( : ν w 1 = ν w 2 = ν u = on. We are interested in non negative solutions of the system. Letting all the other parameters of the model fixed, we shall study the set of the solutions of (3.1 by varying the competition strength β. Let us recall that the assumption (H holds, that is, in this context, λk > µω. We start by recalling a result concerning the regularity of solutions of system (3.1. This result follows from Lemma 2.1, but we report it here for the reader s convenience. Lemma 3.1. Let (w 1, w 2, u H 1 ( be a non negative weak solution to (3.1. Then the solutions are classical. More precisely, (w 1, w 2, u C ( C 2,α ( for any α < 1 and the regularity is limited only by that of ; (3.2 (w 1, w 2, u are non negative and bounded uniformly in β, that is w 1, w 2, u λ/µ u + w 1 + w 2 (λ + ωλ µω and either all the inequalities are strict or the solution is constant;

19 PREY-PREDATORS, PART I 19 Proof. All the assertions in the statement are rather straightforward consequences of the maximum principle and the classical regularity theory of elliptic equations. We only observe that the last inequality follows by summing the three equations together. This yields to (u + w 1 + w 2 (λ + ωu ω(u + w 1 + w 2 (λ + ω λ µ ω(u + w 1 + w 2. We conclude again by virtue of the maximum principle. Lemma 3.1 gives a description of the solutions of the system (3.1, but it contains no information about the existence of such solutions. In the following, our aim is to complete this aspect, showing that the set of solutions is rich. Before doing so, we need to introduce some notation. For a given solution (w 1, w 2, u F( of the system (3.1, the Gâteaux derivate in F( associated to (3.1 computed at (w 1, w 2, u is given by L β ϕ = ϕ A β ϕ, for any ϕ F( where A β = A β (w 1, w 2, u C 2,α (, R 3 3 is ω + ku βw 2 βw 1 kw 1 A = A β = βw 2 ω + ku βw 1 kw 2 ku ku λ 2µu kw 1 kw 2. The solution (w 1, w 2, u is said to be (strongly linearly stable if any non-trivial solution (γ,ϕ of the linearized equation L β ϕ = γϕ has necessarily Re(γ > and weakly stable we can only infer that Re(γ. It is said to be (strongly linearly unstable if, on the contrary, there exists a non-trivial solution with Re(γ <. If the solution (w 1, w 2, u in the previous definition is constant, its stability can be directly deduced from the spectrum of the matrix A β or, more explicitly, from that of A. We start with the simplest scenario, that is the limit case β =. Under this assumption, since the densities of predators do not interact directly with each other, we can simplify drastically the system and give a complete description of the set of solutions of the system.

20 2 HENRI BERESTYCKI AND ALESSANDRO ZILIO Lemma 3.2. The unique non negative solutions (w 1, w 2, u to the system (3.1 with β = are the two unstable constant solutions ( (,,,,, λ µ and the one-parameter family of (weakly stable ones ( λk µω λk µω s [, 1] k 2 s, k 2 (1 s, ω. k Proof. In this proof, we shall only classify the solutions; the study of the stability will be postponed until Lemma 3.3, where we shall address the question about stability of constant solutions for β more generally. Since for β = the densities of predators do not interact directly with each other, we can simplify the system introducing the new variable V = w 1 + w 2, which, together with u is a solution of the classical (i.e. one predator Lotka-Volterra system V = ωv + kvu u = λu µu 2 kvu in in ν V = ν u = on. By a classical result of Mimura [17, Theorem 1] it follows that the previous system has only constant solutions, that is solutions of the algebraic system (ku ωv = (λ µu kvu = When V =, we have the solutions u = or u = λ/µ which correspond to the first two solutions in the statement (recall that w 1 and w 2 are non negative, that is, in this case, w 1 = w 2 =. On the other hand, if u = ω/k, we obtain the solution V = w 1 + w 2 = (λk µω/k 2. Substituting this information in (3.1 we obtain that both w 1 and w 2 are harmonic functions, hence constants. As we shall see later, the value β = corresponds to a bifurcation point of multiplicity one for the system (3.1 around the solution (w 1, w 2, u = ( λk µω 2k 2, λk µω 2k 2, ω, k

21 PREY-PREDATORS, PART I 21 so that the one-parameter family of solutions of Lemma 3.2 is nothing but the branch of solutions emanating from it. Lemma 3.3. When β >, system (3.1 admits four different types of constant solutions: (a the solution (,,, which is strongly unstable; (b the solution which is strongly unstable; (c the solutions w 1 =, w 2 =, u = λ k w 1 = λk µω k 2, w 2 =, u = ω k and w 1 =, w 2 = λk µω k 2, u = ω k which are strongly stable; (d the family of solutions w 1 = w 2 = λk µω λβ + 2kω, u = µβ + 2k2 µβ + 2k 2 which are unstable for β >. In particular, in this latter case, { } λk µω σ(a β = β µβ + 2k 2, γ 1,β, γ 2,β where γ 1,β and γ 2,β are two, possibly complex conjugate, eigenvalues with negative real part. Later we will prove that the solutions in Lemma 3.3 are the only solution of (3.1 when β > is sufficiently small (compare with Proposition Proof. The proof is a rather straightforward computation, but we include it here in order to glean from it an interpretation of the results. The solution (,, corresponds to the matrix ω A β = ω λ which is already in a diagonal form. The instability of this solution is caused by the eigenvalues λ < of A, which corresponds to the constant eigenfunction (,, 1. As a result, in

22 22 HENRI BERESTYCKI AND ALESSANDRO ZILIO complete accordance with other biological models, it implies that a logistic growth law in the prey population is responsible for an (initial exponential growth, uniform in all the domain, at least when the population is small. Let us observe that none of the spectral and stability properties of the trivial solution depends on the competition β. Similar computations hold for the solution (,, λ/µ, whose associated matrix is A β = λk µω µ λk µω µ λk/µ λk/µ λ. Here the eigenvalues are λk µω µ (of multiplicity 2 and λ. The eigenspaces of the matrix is generated by the vectors 1 λk λk µω+λµ, 1 λk λk µω+λµ and 1 respectively. To discuss the solutions of type (c, consider for example the solution w 1 = (λk µω/k 2, w 2 = and u = ω/k. In this case the matrix A β is A β = βw 1 kw 1 βw 1 ω ω µω/k. Thus, the spectrum of A β consists of βw 1, µω/k ± (µω/k 2 4kωw 1, 2 which implies strong stability of these solutions. As already observed in the previous section, this result in unchanged even when the two populations of predators have different parameters, as long as β > (compare Proposition 2.4.

23 PREY-PREDATORS, PART I 23 In the case of the constant solutions of type (d, recalling that here w 1 = w 2, the matrix A β reduces to A β = βw 1 kw 1 βw 1 kw 1 ku ku µu. By direct inspection, we see that βw 1 is an eigenvalue, implying in particular that these solutions are unstable for β >. Using this information, we can factorize the characteristic polynomial of A β, yielding [ ] det(a γid = (γ βw 1 γ 2 + (βw 1 + µuγ + (2k 2 uw 1 + βµuw 1 and the spectrum of A β consists of βw 1, (βw 1 + µu ± (βw 1 + µu 2 (2k 2 uw 1 + βµuw 1, 2 and this concludes the proof. The set of non-trivial constant solutions undergoes a transformation as β changes from β = to β >, see Lemma 3.2. Moreover, the spectrum of the matrix A, computed on the linear set of solutions is given by, µω/k ± (µω/k 2 4kω(w 1 + w 2. 2 The zero eigenvalue underlines the degeneracy of the constant solutions, as they form a linear subspace, while the other two strictly negative eigenvalues confirm that this set of solutions is stable with respect to perturbations that move away from this configuration, i.e. non homogeneous perturbation (see Proposition 2.2. Remark 3.4. The stability of the solutions belonging to the classes (a, (b and (c does not depend on β. More precisely, in the classes (a and (b the spectrum of A β is independent of β, while in the third case (c the spectrum is also contained in C := {z C : Re(z < }. We can say more about constant solution, and in particular we have that if a component is constant, so are the other.

24 24 HENRI BERESTYCKI AND ALESSANDRO ZILIO Lemma 3.5. For a solution (w 1, w 2, u of (3.1, if one component is constant, then also all the other components are constant. Proof. The case for β = is already considered in Lemma 3.2. Thus we can assume β >. We start by assuming that u is constant. If u is zero, we conclude directly by the maximum principle. Assuming that u is a positive constant, from the equation in u, we find that necessarily w 1 + w 2 = λ µu k is a non-negative constant. This yields in the equation for w i, i = 1, 2: ( w i = ω + ku + β µ k u β λ k + βw i w i ν w i = on. Summing up the two equations, we obtain moreover w w2 2 = λ µu k As a result, we have obtained the identities ( λ µu k ku ω β w 1 + w 2 = a, w w2 2 = b for some non-negative constant a and b, which directly implies that w 1 and w 2 are constant. Using this information, it is also possible to compute explicitly the solutions, and in particular we find u = ω/k. We now assume that w 1 is constant. From the equation in w 1 we find that w 1 = or βw 2 = ku ω. The former case is equivalent to assuming β =. We then need only to address the latter. Substituting the previous identity in the equation in u we find u = (A Bu u in ν u = on. where A is a real constant and B is a strictly positive constant. If A then u is zero. On the other hand, if A > by the maximum principle (see Lemma 3.6 below, we find that u is again a constant, and we can conclude as above.

25 PREY-PREDATORS, PART I 25 Note that the arguments of the proof are only valid in the case of two predators w 1, w 2. In the previous result, we made use of the following classical consequence of the maximum principle. Lemma 3.6. Let R n a bounded smooth domain, A and B positive constants. If u H 1 ( is a non negative solution to then u or u A/B. u = (A Buu ν u = in on Here, we are mostly interested in solutions which are not homogeneous in space (i.e., non constant solutions. We will derive their existence through bifurcation arguments. First, we introduce some notation. Definition 3.7. We denote with P R + F( the set of all solutions (β, w 1,β, w 2,β, u β of (3.1 with competition parameter β > such that all of its components are strictly positive. Let also S stand for the set of constant solutions (β, w 1, w 2, u of the form w 1 = w 2 = for all values of β >. Lastly, we let λk µω λβ + 2kω, u = µβ + 2k2 µβ + 2k 2 S 1 = P \ S and S = S 1, where the closure is taken in the R F( topology. Observe that the solutions in S are parameterized in β. We start with the asymptotic analysis when β of the spectrum associated to solutions of type S. Lemma 3.8. Let (w 1, w 2, u S. The eigenvalues of A β behave like ( λ + λk µω λk µω β, γ µβ + 2k2 µ 1,β λk µω µ, γ 2,β as β. As a consequence, the supremum of the spectrum of the matrix A β is described, in terms of β, by the curve β β λk µω µβ + 2k 2.

26 26 HENRI BERESTYCKI AND ALESSANDRO ZILIO Moreover, this supremum of the spectrum is monotone increasing in β and its limit as β can be made arbitrarily large by taking µ small accordingly. In particular, in the limit case µ =, the spectrum is unbounded. Lastly, the unstable direction of A β is spanned by the eigenvector (1, 1,. We can now derive a result concerning the existence of non constant solutions: our construction is implicit and uses the topological degree argument through a bifurcation analysis of the set of constant solutions. Let = γ < γ 1 γ 2... denoted the unbounded sequence of eigenvalues of the Laplace operator with homogeneous Neumann boundary conditions and let {ψ i } be the corresponding eigenfunctions: ψ i = γ i ψ i (3.3 ν ψ i = in on. We define n N to be the largest index corresponding to an eigenvalue γ n such that γ n < λk µω. µ We assume in the following that n 1. Let also β n > be defined by β n λk µω µβ n + 2k 2 = γ n. Observe that n can be made as large as desired by taking µ small accordingly. Theorem 3.9. For any 1 n n, if γ n is an eigenvalue of odd multiplicity, then (β n, w 1,n, w 2,n, u n is a bifurcation point from the branch of solutions S into non constant solutions S. More precisely, there exists a maximal closed and connected subset C n S of solutions of (3.1 such that C n contains the point (β n, w 1,n, w 2,n, u n and either C n is unbounded in β, or C n contains another point (β m, w 1,m, w 2,m, u m for a different value of m. Aside from these bifurcation points emanating from the branch of solutions S, the set C n consists of solutions (w 1, w 2, u which are non constant.

27 PREY-PREDATORS, PART I 27 Remark 3.1. One could wonder what happens for the eigenvalue γ =. Actually, this is already contained in the previous remarks: indeed, γ corresponds to the value β =, and we have already observed in Lemma 3.2 that in this situation there exists a one dimensional subspace of constant solutions emanating from this point. Thus it is also a bifurcation point from S (in such case, the branch is explicit and the solutions are constant. This point is particular and indeed we do not take it into account in the statement of Theorem 3.9. Remark If one assumes some symmetry properties for the domain, one can then also give a more detailed description of the branches in Theorem 3.9. In particular we can show that the symmetries of the eigenfunctions are preserved along a global branch of solutions, see for instance [2, 21]. Proof. The theorem follows from the classical bifurcation theorem of Rabinowitz, see [18, 19]. For β > and a corresponding nontrivial constant solution (w 1, w 2, u with w 1 = w 2, we look for a new solution of the form (w 1 + ϕ 1, w 2 + ϕ 2, u + ϕ, for small perturbations ϕ = (ϕ 1, ϕ 2, ϕ F(. Inserting this ansatz in the system (3.1 we obtain kϕ 1 ϕ βϕ 1 ϕ 2 (3.4 ϕ = A β ϕ + kϕ 2 ϕ βϕ 1 ϕ 2 = A βϕ + H(β,ϕ µϕ 2 k(ϕ 1 + ϕ 2 ϕ in completed by homogeneous Neumann boundary conditions. Here the nonlinear functional H : (R, F( F( is continuous and H(β,ϕ F( C ϕ 2 F( for a constant C > that can be chosen uniformly on compact sets of β [, +. Let us now introduce the operator L K(F(; F( defined as the linear map such that for any u, f F( u + u = f in u = Lf ν u = on. We can rewrite the perturbed system as (3.5 ϕ = (A β + IdLϕ + LH(β,ϕ = (A β + IdLϕ + h(β,ϕ where now h : (R, F( F( is a compact operator. Furthermore, it is such that h(β,ϕ F( C ϕ 2 F( with a constant C > that again can be chosen uniformly on

28 28 HENRI BERESTYCKI AND ALESSANDRO ZILIO compact sets of β. We are now in a position to apply the global bifurcation theorem of Rabinowitz [18, 19]. Indeed, as β varies, (A β + IdL is a homotopy of compact operators. It is known that a sufficient condition for a value β to be a bifurcation point for the equation (3.5 is that the set of solutions to the linear equation ϕ = (A β + IdLϕ has odd dimension. This equation translates into the 3-component system ϕ = A β ϕ in ν ϕ = on. We have already studied the spectral properties of the matrix A β in Lemma 3.3. The matrix has a unique positive eigenvalue β(λk µω/(µ β + 2k 2 that correspond to the eigenvector ( 1, 1,. As a consequence, (γ i, ψ i is an eigenvalue-eigenvector couple of (3.3 and β(λk µω/(µ β + 2k 2 = γ i if and only if ϕ = (ψ i, ψ i, solves the previous system for the prescribed value of β. In particular if γ i has odd multiplicity, then β is a bifurcation point in the sense of the theorem. It remains to show that the continua C n are either unbounded in β or meet the set S in another bifurcation point: recalling the global bifurcation theorem of Rabinowitz [18, Theorem 1.3], we already know that each continuum is either unbounded in R F( or touches the set S at an other bifurcation point. Hence the statement in the theorem is reduced to showing that if the continuum C n is unbounded, then it must be unbounded in β. We recall that, by Lemma 3.1, the non-negative solutions satisfy the inequalities w 1, w 2, u λ/µ, u + w 1 + w 2 (λ + ω2 4µω and either all the inequalities are strict or the solution is constant. It follows that if β is bounded on C n, there must exists on C n a solution which is constant. In view of Lemma 3.3, discarding the solutions on S, the only possibilities are solutions which are either case (a and (b strongly unstable or case (c strongly stable. Recall that these properties do not depend on β >. We shall exclude these possibilities in the following results. Lemma The set of solutions R (,, is isolated in P for β bounded.

29 PREY-PREDATORS, PART I 29 Proof. We assume that there exists a sequence (β n, w 1,n, w 2,n, u n P \ R (,, of solutions of (3.1 such that β n > and, as n +, we have β n β [, + and v n := (w 1,n, w 2,n, u n (,, in F(. We can rewrite (3.1 as ω ku n w 1,n β n w 1,n w 2,n v n = ω v n + kw 2,n u n β n w 1,n w 2,n λ µu 2 n k(w 1,n + w 2,n u n = A,βv n + H (β n, v n in, where H : (R, F( F( is continuous and H (β, v F( C v 2 F( locally at β = β. We follow a reasoning similar to that of Theorem 3.9. We recall that the eigenvalues of the Laplacian with Neumann boundary conditions are non negative. Thus, by the stability analysis of Lemma 3.3, for n + we have that there exists ε n and an eigenpair (γ i, ψ i of (3.3 such that λ = γ i and v n = ε n 1 ψ i + o(ε n. Since λ >, it must be that the index i is strictly positive. But then the eigenfunction ψ i changes sing in, and for n sufficiently large, so does u n. We reach the desired contradiction, as we are considering only solutions that are non negative in. In Lemma 4.5 we will show that the same conclusion holds for β n +, that is, the sets R (,, and P \ R (,, are at a positive distance. We now turn to the other line of constant solutions. Lemma The set of solutions R (,, λ/µ is isolated in P for β bounded. Proof. The proof is rather similar to that of Lemma We omit the details. We only point out that this time the conclusion is reached exploiting the expansion of the solutions as 1 λk µω = γ µ i and v n = ε n ψ i + o(ε n λk λk µω+λµ and the assumption (H, that implies again γ i >.

30 3 HENRI BERESTYCKI AND ALESSANDRO ZILIO Observe that here, in contrast with the case of R (,,, the sets R (,, λ/µ and P \ R (,, λ/µ are at distance. This is due to the presence of the set S. We can strengthen the result in Theorem 3.9, by showing that on all branches of non constant solutions, β is bounded away from zero. More precisely, we have Proposition There exists β > such that the set of solution of (3.1 consists only of constant solutions if β [, β. Proof. Let us assume that there exists a sequence (β n, w 1,n, w 2,n, u n of non constant solutions of (3.1 such that β n > and β n as n +. Up to striking out a subsequence, v n := (w 1,n, w 2,n, u n converges to a constant solution v := ( w 1, w 2, ū with β =. We have already classified these solutions in Lemma 3.2. By the results in Lemmas 3.12 and 3.13, we know that the sequence (w 1,n, w 2,n, u n must converge to a solution in the linear space of solution of Lemma 3.2, that is the segment s [, 1] v s = ( λk µω k 2 s, λk µω k 2 (1 s, ω. k We now prove that they must converge to the solution v 1/2. Indeed, assume that there exists s [, 1] \ {1/2} such that v n v s. Without loss of generality, we can assume that s > 1/2. Thus, for n sufficiently large, we have w 1,n > w 2,n in. We define g n = β n w 1,n w 2,n < in. For n large, we deduce from (3.1 that w 1,n and w 2,n are both distinct solutions of the linear equation w i,n + (ω ku n w i,n = g n ν w i,n = in on. But then, by Fredholm s alternative it must be that the difference of any two distinct solutions is orthogonal to the zero order term g n, that is = an obvious contradiction. g n (w 1,n w 2,n = β n w 1,n w 2,n (w 1,n w 2,n <

31 PREY-PREDATORS, PART I 31 As a result, we have v n v 1/2 in F(. Thus, up to a subsequence, we can write, v n = ( λk µω µβ n + 2k 2, λk µω µβ n + 2k 2, λβ n + 2kω µβ n + 2k 2 + ϕ n where ϕ n = (ϕ 1,n, ϕ 2,n, ϕ n is such that ϕ n in F( as β n. We let ϕ n = ϕ n + ˆϕ n where ϕ n = 1 ϕ n Plugging these relations in system (3.1, we find ˆϕ n = A βn ˆϕ n + A β ϕ n + H(β n,ϕ n ˆϕ n = in on where we used notations similar to those in the proof of Theorem 3.9. In particular, we have H(β n,ϕ n F( C ϕ n 2 F( for a positive constant C. Let us now test the equation against ˆϕ n. We find (3.6 ˆϕ n 2 Aβn ˆϕ n, ˆϕ n = H(β n,ϕ n ˆϕ n We now derive estimates for the two sides of the previous equation. First, recalling the definition of the function H in (3.4, we see that the right hand side of (3.6 is bounded from above by H(β n,ϕ n ˆϕ n C ϕ n F( ˆϕ n 2 for some positive constant C >. On the other hand, by the results in Lemma 3.3 and 3.8 we find λk µω Aβn ˆϕ n, ˆϕ n =βn µβ n + 2k 2 (1, 1, ˆϕ n 2 + Re(γ 1,n e 1,n ˆϕ n 2 + Re(γ 2,n e 2,n ˆϕ n 2 where γ 1,n and γ 1,n are two complex conjugate eigenvalues of A βn with strictly negative real part and e 1,n and e 1,n are the corresponding eigenvectors. More precisely the following holds β n λk µω µβ n + 2k 2 = o n(1 and Re(γ i,n = µω 2k + o n(1 <.